The cyclic sieving phenomenon for faces of cyclic polytopes

A cyclic polytope of dimension d with n vertices is a convex polytope combinato- rially equivalent to the convex hull of n distinct points on a moment curve in R^d. In this paper, we prove the cyclic sieving phenomenon, introduced by Reiner-Stanton-White, for faces of an even-dimensional cyclic polytope, under a group action that cyclically translates the vertices. For odd-dimensional cyclic polytopes, we enumerate the faces that are invariant under an automorphism that reverses the order of the vertices and an automorphism that interchanges the two end vertices, according to the order on the curve. In particular, for n = d + 2, we give instances of the phenomenon under the groups that cyclically translate the odd-positioned and even-positioned vertices, respectively.